Trigonometry Examples

$\frac{\sqrt{1 - \frac{1}{2}}}{2}$

Step 1

Simplify the numerator.

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Step 1.1

Write $1$ as a fraction with a common denominator.

$\frac{\sqrt{\frac{2}{2} - \frac{1}{2}}}{2}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{\sqrt{\frac{2 - 1}{2}}}{2}$

Step 1.3

Subtract $1$ from $2$ .

$\frac{\sqrt{\frac{1}{2}}}{2}$

Step 1.4

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{1}}{\sqrt{2}}}{2}$

Step 1.5

Any root of $1$ is $1$ .

$\frac{\frac{1}{\sqrt{2}}}{2}$

Step 1.6

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}{2}$

Step 1.7

Combine and simplify the denominator.

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Step 1.7.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}}{2}$

Step 1.7.2

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}{2}$

Step 1.7.3

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}{2}$

Step 1.7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}}{2}$

Step 1.7.5

Add $1$ and $1$ .

$\frac{\frac{\sqrt{2}}{{\sqrt{2}}^{2}}}{2}$

Step 1.7.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\frac{\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}{2}$

Step 1.7.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}{2}$

Step 1.7.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\frac{\sqrt{2}}{2^{\frac{2}{2}}}}{2}$

Step 1.7.6.4

Cancel the common factor of $2$ .

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Step 1.7.6.4.1

Cancel the common factor.

$\frac{\frac{\sqrt{2}}{2^{\frac{2}{2}}}}{2}$

Step 1.7.6.4.2

Rewrite the expression.

$\frac{\frac{\sqrt{2}}{2^{1}}}{2}$

Step 1.7.6.5

Evaluate the exponent.

$\frac{\frac{\sqrt{2}}{2}}{2}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$

Step 3

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 3.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

$\frac{\sqrt{2}}{2 \cdot 2}$

Step 3.2

Multiply $2$ by $2$ .

$\frac{\sqrt{2}}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt{2}}{4}$

Decimal Form:

$0.35355339 \dots$